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Iterative process for a strictly pseudocontractive mapping in uniformly convex Banach spaces
Journal of Inequalities and Applications volume 2014, Article number: 377 (2014)
Abstract
This paper is concerned with a new method to prove the weak convergence of a strictly pseudocontractive mapping in a puniformly convex Banach space with more relaxed restrictions on the parameters. Our results extend and improve the corresponding earlier results.
MSC:41A65, 47H17, 47J20.
1 Introduction and preliminaries
In 1967, Browder and Petryshyn [1] gave the classical definition for strictly pseudocontractive mappings in Hilbert spaces for the first time.
Definition 1.1 Let C be a nonempty closed convex subset of a real Hilbert space H. T:C\to H is called a BrowderPetryshyntype kstrictly pseudocontractive mapping. Then there exists k\in [0,1) such that for every x,y\in C
In 2010, Zhou [2] gave a new definition for kstrictly pseudocontractive mappings in quniformly smooth Banach spaces.
Definition 1.2 Let C be a nonempty closed convex subset of a quniformly smooth Banach space X. T:C\to C is called a Zhoutype kstrictly pseudocontractive mapping, if there exists k\in [0,1) such that for every x,y\in C
In 2009, Hu and Wang [3] gave another definition for kstrictly pseudocontractive mappings in puniformly convex Banach spaces.
Definition 1.3 Let C be a nonempty closed convex subset of a puniformly convex Banach space X. T:C\to C is called a Hutype kstrictly pseudocontractive mapping, if there exists k\in [0,1) such that for every x,y\in C
Remark 1.1 The mappings defined by (1.1) and (1.2) are pseudocontractive mappings, but the mapping defined by (1.3) may not be pseudocontractive in general Banach spaces.
Remark 1.2 If and only if q=2, the mappings defined by (1.1) and (1.2) are equivalent.
Remark 1.3 If p=q=2, the mappings defined by (1.1), (1.2), and (1.3) are equivalent in Hilbert space.
In 1979, Reich [4] established a weak convergence theorem via a Manntype iterative process for nonexpansive mapping in a uniformly convex Banach space with Fréchet differentiable norm.
Theorem R Let C be a closed convex subset of a uniformly convex Banach space X with a Fréchet differentiable norm and T:C\to C a nonexpansive mapping with F(T)\ne \mathrm{\varnothing}. For any {x}_{1}\in C, the iterative sequence \{{x}_{n}\} is defined by {x}_{n+1}=(1{\alpha}_{n}){x}_{n}+{\alpha}_{n}T{x}_{n}, where the real sequence \{{\alpha}_{n}\}\subset [0,1] and {\sum}_{n=1}^{\mathrm{\infty}}(1{\alpha}_{n}){\alpha}_{n}=\mathrm{\infty}. Then the sequence \{{x}_{n}\} converges weakly to a fixed point of T.
In 2007, Marino and Xu [5] improved Reich’s [4] result and gave several weak convergence theorems via the normal Mann iterative algorithm for strictly pseudocontractive mappings in Hilbert spaces. Further, they proposed an open problem: Do the main results of [5]still hold true in the framework of Banach spaces which are uniformly convex and have a Fréchet differentiable norm?
In 2009, Hu and Wang [3] considered above problem in a puniformly convex Banach space and established the following theorem.
Theorem H Let C be a closed convex subset of a puniformly convex Banach space X with a Fréchet differentiable norm and T:C\to C be a kstrictly pseudocontractive mapping in the light of (1.3) with coefficients p,k<min\{1,{2}^{(p2)}{c}_{p}\} and F(T)\ne \mathrm{\varnothing}. For any {x}_{1}\in C and n>1, the iterative sequence \{{x}_{n}\} is defined by {x}_{n+1}=(1{\alpha}_{n}){x}_{n}+{\alpha}_{n}T{x}_{n}, where the real sequence \{{\alpha}_{n}\}\subset [0,1] and 0<\epsilon \le {\alpha}_{n}\le 1\epsilon <1\frac{{2}^{p2}k}{{c}_{p}}. Then the sequence \{{x}_{n}\} converges weakly to a fixed point of T.
Question Can one relax the restriction on the parameters {\alpha}_{n} in Theorem H and simplify its proof?
The purpose of this paper is to solve the question mentioned above. To prove our results, we need the following lemmas.
Lemma 1.1 (see [3])
Let C be a nonempty closed convex subset of a puniformly convex Banach space X and T:C\to C be a Hutype strictly pseudocontractive mapping in the light of (1.3). For \alpha \in (0,1), define {T}_{\alpha}:C\to C by {T}_{\alpha}=(1\alpha )x+\alpha Tx, for x\in C. If \alpha \in (0,1(k{2}^{p2})/{c}_{p}), then {T}_{\alpha} is a nonexpansive mapping and F({T}_{\alpha})=F(T).
Lemma 1.2 (see [3])
Let C be a nonempty closed convex subset of a puniformly convex Banach space X and T:C\to C be a Hutype strictly pseudocontractive mapping in the light of (1.3). For \mu \in (0,1), {T}_{\mu}:C\to C is defined by {T}_{\mu}=(1\mu )x+\mu Tx, for x\in C. Then the following inequality holds:
where {W}_{p}(\mu )={\mu}^{p}(1\mu )+\mu {(1\mu )}^{p}.
Lemma 1.3 (see [6])
Let C be a nonempty closed convex subset of a puniformly convex Banach space X and T:C\to C be a nonexpansive mapping, then IT is demiclosed at zero.
Lemma 1.4 (see [7])
Let C be a nonempty closed convex subset of a puniformly convex Banach space X which satisfies the Opial condition and T:C\to C be a quasinonexpansive mapping with F(T)\ne \mathrm{\varnothing}. If IT is demiclosed at zero, then for any {x}_{0}\in C, the normal Mann iteration \{{x}_{n}\} defined by
converges weakly to a fixed point of T, where \{{\alpha}_{n}\}\subset [0,1] and {\sum}_{n=0}^{\mathrm{\infty}}min\{{\alpha}_{n},(1{\alpha}_{n})\}=\mathrm{\infty}.
Lemma 1.5 (see [7])
Let C be a nonempty closed convex subset of a puniformly convex Banach space X whose dual space {X}^{\ast} satisfies KadecKlee property and T:C\to C be a nonexpansive mapping with F(T)\ne \mathrm{\varnothing}. Then, for any {x}_{0}\in C, the normal Mann iteration \{{x}_{n}\} defined by
converges weakly to a fixed point of T, where \{{\alpha}_{n}\}\subset [0,1] and {\sum}_{n=0}^{\mathrm{\infty}}min\{{\alpha}_{n},(1{\alpha}_{n})\}=\mathrm{\infty}.
Now we are in a position to state and prove the main results in this paper.
2 Main results
Theorem 2.1 Let C be a nonempty closed convex subset of a puniformly convex Banach space X with Fréchet differential norm. Let T:C\to C be a Hutype kstrictly pseudocontractive mapping in the light of (1.3) with coefficients p,k<min\{1,{2}^{(p2)}{c}_{n}\} and F(T)\ne \mathrm{\varnothing}. Assume that a real sequence \{{\alpha}_{n}\} in [0,1] satisfies the conditions:

(i)
0\le {\alpha}_{n}\le \alpha =1(k{2}^{p2}/{c}_{p}), n\ge 0;

(ii)
{\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}[(1{\alpha}_{n}){2}^{2p}{c}_{p}k]=\mathrm{\infty}.
For any {x}_{0}\in C, the normal Mann iterative sequence \{{x}_{n}\} is defined by
Then the sequence \{{x}_{n}\} defined by (2.1) converges weakly to a fixed point of T.
Proof Let {T}_{\alpha} be given as in Lemma 1.1. Then {T}_{\alpha}:C\to C is a nonexpansive mapping with F({T}_{\alpha})=F(T). Set {\beta}_{n}=\frac{\alpha {\alpha}_{n}}{\alpha}. Then (2.1) reduces to {x}_{n+1}={\beta}_{n}{x}_{n}+(1{\beta}_{n}){T}_{\alpha}{x}_{n}.
We note that
By using Theorem R, we conclude that \{{x}_{n}\} converges weakly to a fixed point of {T}_{\alpha}, and of T. The proof is complete. □
Remark 2.2 Theorem 2.1 relaxes the iterative parameters in Theorem H and our proof method is also quite concise.
Theorem 2.3 Let C be a nonempty closed convex subset of a puniformly convex Banach space X which satisfies the Opial condition. Let T:C\to C be a Hutype kstrictly pseudocontractive mapping in the light of (1.3) with coefficients p,k<min\{1,{2}^{(p2)}{c}_{p}\} and F(T)\ne \mathrm{\varnothing}. Assume that the real sequence \{{\alpha}_{n}\} in [0,1] satisfies the conditions:

(i)
0\le {\alpha}_{n}\le \alpha =1(k{2}^{p2}/{c}_{p}), n\ge 0;

(ii)
{\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}[(1{\alpha}_{n}){2}^{2p}{c}_{p}k]=\mathrm{\infty}.
For any {x}_{0}\in C, the normal Mann iteration \{{x}_{n}\} is defined by
Then the sequence \{{x}_{n}\} defined by (2.2) converges weakly to the fixed point of T.
Proof Let {T}_{\alpha} be given as in Lemma 1.1. Then {T}_{\alpha}:C\to C is a nonexpansive mapping with F({T}_{\alpha})=F(T). Set {\beta}_{n}=\frac{\alpha {\alpha}_{n}}{\alpha}. Then (2.2) reduces to {x}_{n+1}={\beta}_{n}{x}_{n}+(1{\beta}_{n}){T}_{\alpha}{x}_{n}. As shown in Theorem 2.1, {\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}(1{\beta}_{n})=\mathrm{\infty}. By Lemma 1.3, I{T}_{\alpha} is demiclosed at zero. By Lemma 1.4, we conclude that \{{x}_{n}\} converges weakly to a fixed point of {T}_{\alpha}, and of T. The proof is complete. □
Theorem 2.4 Let C be a nonempty closed convex subset of a puniformly convex Banach space X with the dual space {X}^{\ast} satisfying the KadecKlee property. Let T:C\to C be a Hutype kstrictly pseudocontractive mapping in the light of (1.3) with coefficients p,k<min\{1,{2}^{(p2)}{c}_{p}\} and F(T)\ne \mathrm{\varnothing}. Assume that the real sequence \{{\alpha}_{n}\} in [0,1] satisfies the conditions:

(i)
0\le {\alpha}_{n}\le \alpha =1(k{2}^{p2}/{c}_{p}), n\ge 0;

(ii)
{\sum}_{n=0}^{\mathrm{\infty}}{\alpha}_{n}[(1{\alpha}_{n}){2}^{2p}{c}_{p}k]=\mathrm{\infty}.
For any {x}_{0}\in C, the normal Mann iteration \{{x}_{n}\} is defined by
Then the sequence \{{x}_{n}\} defined by (2.3) converges weakly to a fixed point of T.
Proof Let {T}_{\alpha} be given as in Lemma 1.1. Then {T}_{\alpha}:C\to C is a nonexpansive mapping with F({T}_{\alpha})=F(T). Set {\beta}_{n}=\frac{\alpha {\alpha}_{n}}{\alpha}. Then (2.3) reduces to {x}_{n+1}={\beta}_{n}{x}_{n}+(1{\beta}_{n}){T}_{\alpha}{x}_{n}. As shown in Theorem 2.1, {\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}(1{\beta}_{n})=\mathrm{\infty}. By using Lemma 1.5, \{{x}_{n}\} defined by (2.3) converges weakly to a fixed point of {T}_{\alpha}, and of T. The proof is complete. □
References
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Acknowledgements
This research was supported by the National Natural Science Foundation of China (11071053).
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Zhou, Y., Zhou, H. & Wang, P. Iterative process for a strictly pseudocontractive mapping in uniformly convex Banach spaces. J Inequal Appl 2014, 377 (2014). https://doi.org/10.1186/1029242X2014377
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DOI: https://doi.org/10.1186/1029242X2014377
Keywords
 strictly pseudocontractive mapping
 nonexpansive mapping
 Mann iteration
 uniformly convex Banach space